A simple interpretation of the absence of quantum diffusion for one- and two-dimensional disorder

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Abstract
Using an elementary approach, it is shown that large time diffusive propagation of wave intensity in one- and two-dimensional disordered systems is incompatible with the sort of energy spectrum (local density of states) one associates with extended states. All eigenstates are therefore localised however weak the disorder, and there is no diffusion over large distances and time scales. This result is a simple consequence of Polya's theorem, which states that random walks in one and two dimensions always return eventually to the neighbourhood of their origin. Estimates of the localisation length are obtained which agree with results from more sophisticated methods: for example, recently proposed scaling theories.

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